General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

We derive general bounds for the large time size of supnorm values of solutions to one-dimensional advection-diffusion equations with initial data for some and arbitrary bounded advection speeds , introducing new techniques based on suitable energy arguments. Some open problems and related results are also given. 1. Introduction In this work, we obtain very general large time estimates for supnorm values of solutions to parabolic initial value problems of the form for arbitrary continuously differentiable advection fields . Here, by solution to (1a) and (1b) in some time interval , , we mean a function which is bounded in each strip , , solves (1a) in the classical sense for , and satisfies in as . It follows from the a priori estimates given in Section 2 that all solutions of problem (1a), (1b) are actually globally defined , with for each finite. Given , what then can be said about the size of supnorm values for ? When for all , it is well known that, for each , is monotonically decreasing in , with for some constant that depends only on ; see, for example, [1–5]. For general , however, estimating is much harder. To see why, let us illustrate with the important case , where one has as recalled in Theorem 1. Writing (1a) as we observe on the right hand side of (4) that is pushed to grow at points where . If this condition persists long enough, large values of might be generated, particularly at sites where . Now, because of constraint (3), any persistent growth in solution size will eventually create long thin structures as shown in Figure 1, which, in turn, tend to be effectively dissipated by viscosity. The final overall behavior that ultimately results from such competition is not immediately clear, either on physical or on mathematical grounds. Figure 1: Solution profiles showing typical growth in regions with , where . After reaching maximum height, solution starts decaying very slowly due to its spreading and mass conservation (decay rate is not presently known). As shown in (4), it is not the magnitude of itself but instead its oscillation that is relevant in determining . Accordingly, we introduce the quantity defined by which plays a fundamental role in the analysis. Our main result is now easily stated. Main Theorem. For each , one has1 where . In particular, in the important case considered above, we obtain, using (3), so that stays uniformly bounded for all time in this case.2 Estimates similar to (6) can also be shown to hold for the -dimensional problem but to simplify our discussion we consider here the case only. Our derivation of (6),